Prediction of an Outcome Using Trajectories Estimated from a Linear Mixed Model

Abstract

In longitudinal data, interest is usually focused on the repeatedly measured variable itself. In some situations, however, the pattern of variation of the variable over time may contain information about a separate outcome variable. In such situations, longitudinal data provide an opportunity to develop predictive models for future observations of the separate outcome variable given the current data for an individual. In particular, longitudinally changing patterns of repeated measurements of a variable measured up to time t , or trajectories, can be used to predict an outcome measure or event that occurs after time t .

In this article, we propose a method for predicting an outcome variable based on a generalized linear model, specifically, a logistic regression model, the covariates of which are variables that characterize the trajectory of an individual. Since the trajectory of an individual contains estimation error, the proposed logistic regression model constitutes a measurement error model. The model is fitted in two steps. First, a linear mixed model is fitted to the longitudinal data to estimate the random effect that characterizes the trajectory for each individual while adjusting for other covariates. In the second step, a conditional likelihood approach is applied to account for the estimation error in the trajectory. Prediction of an outcome variable is based on the logistic regression model in the second step. The receiver operating characteristic curve is used to compare the discrimination ability of a model with trajectories to one without trajectories as covariates. A simulation study is used to assess the performance of the proposed method, and the method is applied to clinical trial data.

Key Words:

• Generalized linear model
• Linear mixed model
• Longitudinal data
• Prediction
• ROC curve
• Trajectory

ACKNOWLEDGMENT

The authors thank Dr. Stephen Lagakos at the Harvard School of Public Health for his insightful comments.

Notes

¹Including covariates in Eq. (5).

²Not including covariates in Eq. (5).

³SE, standard error.

⁴95% cov., 95% confidence interval coverage probabilities.